Residually finite group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This is a variation of finiteness (groups)|Find other variations of finiteness (groups) |

Definition

Symbol-free definition

A group is said to be finitely approximable or residually finite if it satisfies the following equivalent conditions:

• For any non-identity element of the group, there is a subgroup of finite index not containing that element
• For any non-identity element of the group, there is a normal subgroup of finite index not containing that element
• The intersection of normal subgroups of finite index in it is trivial
• The natural map from the group to its profinite completion, is injective

Formalisms

In terms of the residually operator

This property is obtained by applying the residually operator to the property: finite group
View other properties obtained by applying the residually operator

The group property of being residually finite is obtained by applying the residually operator to the group property of being finite.